# Jacob Wolfowitz
**Jacob Wolfowitz** (19 March 1910 – 16 July 1981) was an American statistician and information theorist, born in Warsaw, who is credited with the Wald–Wolfowitz runs test, a proof of the optimality of Wald's sequential probability ratio test, and the monograph *Coding Theorems of Information Theory*. He was professor of mathematics at Cornell University from 1951 to 1970, then at the [University of Illinois Urbana-Champaign](https://www.edgepedia.org/university-of-illinois-urbana-champaign) from 1970 to 1978, and finished his career as distinguished professor at the [University of South Florida](https://www.edgepedia.org/university-of-south-florida).[1](https://ecommons.cornell.edu/server/api/core/bitstreams/4062042e-9ac9-4dbd-b3ff-91c02d9989d9/content) • [2](https://mathshistory.st-andrews.ac.uk/DSB/Wolfowitz.pdf) • [3](https://magazine.amstat.org/blog/2010/09/01/wolfowitz9_10/)
| Fact | Detail |
|---|---|
| Born – died | 19 March 1910, Warsaw – 16 July 1981, Tampa, Florida[2](https://mathshistory.st-andrews.ac.uk/DSB/Wolfowitz.pdf) |
| Training | PhD in mathematics, New York University, 1942; de facto thesis sponsor Abraham Wald[2](https://mathshistory.st-andrews.ac.uk/DSB/Wolfowitz.pdf) • [1](https://ecommons.cornell.edu/server/api/core/bitstreams/4062042e-9ac9-4dbd-b3ff-91c02d9989d9/content) |
| Career | UNC (1945), Columbia (1946), Cornell (1951–1970), Illinois (1970–1978), South Florida (1978–1981)[3](https://magazine.amstat.org/blog/2010/09/01/wolfowitz9_10/) |
| Signature work | *Coding Theorems of Information Theory* (1961; 3rd ed. 1978); Wald–Wolfowitz runs test (1940)[2](https://mathshistory.st-andrews.ac.uk/DSB/Wolfowitz.pdf) • [4](https://www.oxfordreference.com/display/10.1093/oi/authority.20110803120413951) |
| SPRT optimality | 1948 proof that the sequential probability ratio test minimizes expected sample size[1](https://ecommons.cornell.edu/server/api/core/bitstreams/4062042e-9ac9-4dbd-b3ff-91c02d9989d9/content) |
| Terminology | First use of the word "nonparametric", in a 1942 paper[2](https://mathshistory.st-andrews.ac.uk/DSB/Wolfowitz.pdf) |
| Highest honor | Claude E. Shannon Award, IEEE Information Theory Society, 1979[5](https://www.itsoc.org/profile/8975) |
## Life and career
According to *Amstat News*, Wolfowitz came to the United States with his family in 1920, joining his father in New York, and graduated from the College of the City of New York in 1931.[3](https://magazine.amstat.org/blog/2010/09/01/wolfowitz9_10/) He took an MA at Columbia University in 1933 and, while supporting himself as a high-school teacher, earned the PhD in mathematics from New York University in 1942.[2](https://mathshistory.st-andrews.ac.uk/DSB/Wolfowitz.pdf) In 1934 he married Lillian Dundes, to whom he remained married for the rest of his life.[3](https://magazine.amstat.org/blog/2010/09/01/wolfowitz9_10/)
His academic path ran through the [University of North Carolina](https://www.edgepedia.org/university-of-north-carolina), where he became associate professor in 1945, and Columbia University from 1946, before he joined the mathematics department at Cornell as professor in 1951.[3](https://magazine.amstat.org/blog/2010/09/01/wolfowitz9_10/) In 1970 he moved to the University of Illinois at Urbana-Champaign, retiring in 1978 and becoming distinguished professor at the University of South Florida at Tampa, a post he held until his death.[3](https://magazine.amstat.org/blog/2010/09/01/wolfowitz9_10/) • [7](https://www.upi.com/Archives/1981/07/16/Dr-Jacob-Wolfowitz-a-noted-teacher-and-researcher-in/1628364104000/) He also held visiting appointments at UCLA (1952), the University of Illinois (1953), the Technion in Israel (1957), the Technion, and the [University of Paris](https://www.edgepedia.org/university-of-paris) (1967), and the University of Heidelberg (1969).[8](https://mathshistory.st-andrews.ac.uk/Biographies/Wolfowitz/)
## Sequential analysis and the Wald–Wolfowitz runs test
According to Lionel Weiss's entry on Wolfowitz in the *Dictionary of Scientific Biography*, Wolfowitz met [Abraham Wald](https://www.edgepedia.org/abraham-wald) soon after Wald arrived in New York in the autumn of 1938, and the two quickly became close friends and collaborators; their first joint paper, "Confidence Limits for Continuous Distributions," appeared in the *Annals of Mathematical Statistics* in 1939.[2](https://mathshistory.st-andrews.ac.uk/DSB/Wolfowitz.pdf) • [3](https://magazine.amstat.org/blog/2010/09/01/wolfowitz9_10/) In 1940 the pair published the runs test: a nonparametric, distribution-free test of the randomness of an ordered sequence of binary observations, in which the number of runs expected by chance is calculated and compared with the number observed.[4](https://www.oxfordreference.com/display/10.1093/oi/authority.20110803120413951) A 1942 paper by Wolfowitz contains the first appearance of the word "nonparametric."[2](https://mathshistory.st-andrews.ac.uk/DSB/Wolfowitz.pdf)
During the war he left high-school teaching to join the Statistical Research Group at Columbia, formed in 1942 for war-related research, where he worked alongside Wald until 1945.[1](https://ecommons.cornell.edu/server/api/core/bitstreams/4062042e-9ac9-4dbd-b3ff-91c02d9989d9/content) There Wald developed sequential analysis and conjectured that the sequential probability ratio test (SPRT) minimizes the average number of observations required; no rigorous proof existed until the 1948 joint paper "Optimal Character of the Sequential Probability Ratio Test," which showed that an SPRT with prescribed type I and type II error probabilities minimizes the expected sample size. It may have been the paper of which Wolfowitz was proudest.[1](https://ecommons.cornell.edu/server/api/core/bitstreams/4062042e-9ac9-4dbd-b3ff-91c02d9989d9/content) • [2](https://mathshistory.st-andrews.ac.uk/DSB/Wolfowitz.pdf) • [9](https://nap.nationalacademies.org/nap-cgi/skimchap.cgi?chap=372%E2%80%93384&recid=10683)
## Coding Theorems of Information Theory
From 1957 Wolfowitz devoted a growing share of his research to information theory, and at Cornell in 1961 he wrote the first edition of *Coding Theorems of Information Theory*, which contains most of his results on limits of transmission rates for various types of channels.[1](https://ecommons.cornell.edu/server/api/core/bitstreams/4062042e-9ac9-4dbd-b3ff-91c02d9989d9/content) • [2](https://mathshistory.st-andrews.ac.uk/DSB/Wolfowitz.pdf) The third edition appeared in 1978 from Springer-Verlag, 173 pages with bibliography and index, and has been described as a classic and the only book concentrating on the statistical and probabilistic aspects of noisy-channel communication theory.[8](https://mathshistory.st-andrews.ac.uk/Biographies/Wolfowitz/) • [10](https://archive.org/details/codingtheoremsof0000wolf) Later editions replaced a chapter with one on arbitrarily varying channels and added material on rate distortion, source coding, multiple access channels, and degraded broadcast channels.[11](https://doi.org/10.1016/0016-0032(62)90913-4)
## Later statistical work
Wolfowitz's investigation of why the maximum likelihood estimator works as well as it does began in 1949 and continued for most of his life; a 1956 *Annals of Mathematical Statistics* paper gave what appears to be the first practical example in which maximum likelihood breaks down, in the setting of infinitely many incidental parameters.[3](https://magazine.amstat.org/blog/2010/09/01/wolfowitz9_10/) This line of work led in 1969 to maximum probability estimators, and culminated in the 1974 book *Maximum Probability Estimation and Related Topics*.[9](https://nap.nationalacademies.org/nap-cgi/skimchap.cgi?chap=372%E2%80%93384&recid=10683) • [1](https://ecommons.cornell.edu/server/api/core/bitstreams/4062042e-9ac9-4dbd-b3ff-91c02d9989d9/content) Papers on the inventory problem published in *Econometrica* in April 1952, July 1952, and October 1953 are considered pioneering in inventory theory and an early example of dynamic programming.[3](https://magazine.amstat.org/blog/2010/09/01/wolfowitz9_10/) • [12](http://www.oxfordreference.com/viewbydoi/10.1093/oi/authority.20110803124349722)
### Representative work
- [Asymptotically minimax estimation of concave and convex distribution functions](https://doi.org/10.1007/bf00532690), *Probability Theory and Related Fields*, 1976, 34(1):73–85.[13](https://doi.org/10.1109/tit.1982.1056541)
- [Codes within codes](https://doi.org/10.1007/bf00538118), *Probability Theory and Related Fields*, 1979, 46(3):307–315.[13](https://doi.org/10.1109/tit.1982.1056541)
## Honors and recognition
The IEEE Information Theory Society records Wolfowitz as the 1979 recipient of the Claude E. Shannon Award.[5](https://www.itsoc.org/profile/8975) He was a fellow of the American Academy of Arts and Sciences, the Institute of Mathematical Statistics, and the Econometric Society, a member of the International Statistics Institute, served a term as president of the IMS, was its Rietz and Wald lecturer, held a [Guggenheim Fellowship](https://www.edgepedia.org/guggenheim-fellowship) in 1966–67, and made pioneering contributions to dynamic programming.[1](https://ecommons.cornell.edu/server/api/core/bitstreams/4062042e-9ac9-4dbd-b3ff-91c02d9989d9/content) • [6](https://docslib.org/doc/819956/jacob-wolfowitz-1910-1981) • [12](http://www.oxfordreference.com/viewbydoi/10.1093/oi/authority.20110803124349722)
## Wald, mentorship and legacy
Wald was both collaborator and mentor: the two published joint papers in 1939 and 1940, before Wolfowitz had his PhD, and Wald was the de facto sponsor of Wolfowitz's thesis.[1](https://ecommons.cornell.edu/server/api/core/bitstreams/4062042e-9ac9-4dbd-b3ff-91c02d9989d9/content) After Wald was killed in a plane crash in 1950, a large photograph of Wald occupied a prominent place on Wolfowitz's office wall, and his death removed the strongest tie binding Wolfowitz to Columbia, preceding the 1951 move to Cornell.[3](https://magazine.amstat.org/blog/2010/09/01/wolfowitz9_10/) • [2](https://mathshistory.st-andrews.ac.uk/DSB/Wolfowitz.pdf)
According to a UPI obituary, Wolfowitz died of a heart attack at University Community Hospital in Tampa on 16 July 1981, aged 71.[7](https://www.upi.com/Archives/1981/07/16/Dr-Jacob-Wolfowitz-a-noted-teacher-and-researcher-in/1628364104000/) *Jacob Wolfowitz: Selected Papers* (Springer, 1980) contains a complete bibliography of his 120 publications and the full text of 49 papers he chose as among his most important; an obituary with bibliography appeared in *IEEE Transactions on Information Theory* in 1982 (volume 28, pages 687–690).[2](https://mathshistory.st-andrews.ac.uk/DSB/Wolfowitz.pdf) • [3](https://magazine.amstat.org/blog/2010/09/01/wolfowitz9_10/) His research spanned nonparametric inference, sequential analysis, statistical decision theory, asymptotic statistical theory, design of experiments, queuing and inventory theory, and information theory.[8](https://mathshistory.st-andrews.ac.uk/Biographies/Wolfowitz/)
## References
1. [In Memoriam: Jacob Wolfowitz (Cornell memorial notice)](https://ecommons.cornell.edu/server/api/core/bitstreams/4062042e-9ac9-4dbd-b3ff-91c02d9989d9/content)
2. [Wolfowitz, Jacob, Dictionary of Scientific Biography, via MacTutor](https://mathshistory.st-andrews.ac.uk/DSB/Wolfowitz.pdf)
3. [Jack Wolfowitz: Applying Mathematical Statistics to Practical Problems (Amstat News)](https://magazine.amstat.org/blog/2010/09/01/wolfowitz9_10/)
4. [Wald–Wolfowitz runs test, Oxford Reference](https://www.oxfordreference.com/display/10.1093/oi/authority.20110803120413951)
5. [Member profile | IEEE Information Theory Society](https://www.itsoc.org/profile/8975)
6. [Jacob Wolfowitz 1910–1981 (NAS Biographical Memoir)](https://docslib.org/doc/819956/jacob-wolfowitz-1910-1981)
7. [UPI obituary, July 16, 1981](https://www.upi.com/Archives/1981/07/16/Dr-Jacob-Wolfowitz-a-noted-teacher-and-researcher-in/1628364104000/)
8. [Jacob Wolfowitz, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Biographies/Wolfowitz/)
9. [Biographical Memoir: Jacob Wolfowitz (National Academy of Sciences)](https://nap.nationalacademies.org/nap-cgi/skimchap.cgi?chap=372%E2%80%93384&recid=10683)
10. [Coding theorems of information theory, Internet Archive](https://archive.org/details/codingtheoremsof0000wolf)
11. https://doi.org/10.1016/0016-0032(62)90913-4
12. [Wolfowitz, Jacob, A Dictionary of Statistics (Oxford Reference)](http://www.oxfordreference.com/viewbydoi/10.1093/oi/authority.20110803124349722)
13. [In Memoriam: Jacob Wolfowitz, IEEE Transactions on Information Theory (1982)](https://doi.org/10.1109/tit.1982.1056541)
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